June  2021, 26(6): 2941-2956. doi: 10.3934/dcdsb.2020214

Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials

School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, 519082, China

* Corresponding author: Yulin Zhao

Received  July 2019 Published  June 2021 Early access  July 2020

Fund Project: This research is supported by the NSF of China (No.11971495 and No.11801582)

This paper is devoted to the complete classification of global phase portraits for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Such system is affinely equivalent to one of five normal forms by an algebraic classification of its infinite singular points. Then, we classify the global phase portraits of these normal forms on the Poincaré disc. There are exactly $ 13 $ different global topological structures on the Poincaré disc. Finally we provide the bifurcation diagrams for the corresponding global phase portraits.

Citation: Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214
References:
[1]

M. ÁlvarezA. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416.

[2]

A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators: Adiwes International Series in Physics, vol. 4, Elsevier, 2013.

[3]

J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95.  doi: 10.1006/jdeq.1994.1004.

[4]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields. Ⅰ, Nonlinear Anal., 29 (1997), 783-811.  doi: 10.1016/S0362-546X(96)00088-0.

[5]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields Ⅱ, Nonlinear Anal., 39 (2000), 351-363.  doi: 10.1016/S0362-546X(98)00177-1.

[6]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅲ), Qual. Theory Dyn. Syst., 10 (2011), 203-246.  doi: 10.1007/s12346-011-0052-y.

[7]

A. CimaA. Gasull and F. Mañosas, On polynomial Hamiltonian planar vector fields, J. Differential Equations, 106 (1993), 367-383.  doi: 10.1006/jdeq.1993.1112.

[8]

A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.  doi: 10.1016/0022-247X(90)90359-N.

[9]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661.  doi: 10.1016/j.jde.2014.05.024.

[10]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Adv. Math., 259 (2014), 655-687.  doi: 10.1016/j.aim.2014.04.002.

[11]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 258 (2015), 846-879.  doi: 10.1016/j.jde.2014.10.006.

[12]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 262 (2017), 5518-5533.  doi: 10.1016/j.jde.2017.02.001.

[13]

F. S. DiasJ. Llibre and C. Valls, Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers, Math. Comput. Simulation, 144 (2018), 60-77.  doi: 10.1016/j.matcom.2017.06.002.

[14]

F. Dumortier, Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations, in Bifurcations and Periodic Orbits of Vector Fields, Springer, 1993, 19–73.

[15]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[16]

A. GasullA. Guillamon and V. Mañosa, Phase portrait of Hamiltonian systems with homogeneous nonlinearities, Nonlinear Anal., 42 (2000), 679-707.  doi: 10.1016/S0362-546X(99)00131-5.

[17]

H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass, 1951.

[18]

A. Guillamon and C. Pantazi, Phase portraits of separable Hamiltonian systems, Nonlinear Anal., 74 (2011), 4012-4035.  doi: 10.1016/j.na.2011.03.030.

[19]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[20]

J. S. W. Lamb and M. Roberts, Reversible equivariant linear systems, J. Differential Equations, 159 (1999), 239-279.  doi: 10.1006/jdeq.1999.3632.

[21]

H. LiangJ. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681.  doi: 10.1007/s11071-014-1541-8.

[22]

J. LlibreY. P. Martínez and C. Vidal, Phase portraits of linear type centers of polynomial hamiltonian systems with hamiltonian function of degree $5$ of the form $H = H_1(x)+ H_2(y)$, Discrete Contin. Dyn. Syst. Ser., 39 (2019), 75-113.  doi: 10.3934/dcds.2019004.

[23]

J. LlibreY. P. Martínez and C. Vidal, Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the $y$-axis, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 887-912.  doi: 10.3934/dcdsb.2018047.

[24]

J. LlibreR. Oliveira and C. Valls, Phase portraits for some symmetric Riccati cubic polynomial differential equations, Topology Appl., 234 (2018), 220-237.  doi: 10.1016/j.topol.2017.11.023.

[25]

J. Llibre and C. Pessoa, Phase portraits for quadratic homogeneous polynomial vector fields on $\Bbb S^2$, Rend. Circ. Mat. Palermo, 58 (2009), 361-406.  doi: 10.1007/s12215-009-0030-2.

[26]

N. Minorsky, Nonlinear Oscillations, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962.

[27]

D. A. Neumann, Classification of continuous flows on $2$-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.  doi: 10.1090/S0002-9939-1975-0356138-6.

[28]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (i), Journal de Mathématiques Pures et Appliquées, 7 (1881), 375–422.

[29]

B. Qiu and H. Liang, Classification of global phase portrait of planar quintic quasi-homogeneous coprime polynomial systems, Qual. Theory Dyn. Syst., 16 (2017), 417-451.  doi: 10.1007/s12346-016-0199-7.

[30]

J. Reyn, Phase Portraits of Planar Quadratic Systems, vol. 583, Springer, New York, 2007.

[31]

D. Schlomiuk and X. Zhang, Quadratic differential systems with complex conjugate invariant lines meeting at a finite point, J. Differential Equations, 265 (2018), 3650-3684.  doi: 10.1016/j.jde.2018.05.014.

[32]

Y. Tian and Y. Zhao, Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the $y$-axis, Nonlinear Anal., 192 (2020), 111658, 27pp. doi: 10.1016/j.na.2019.111658.

[33]

X. Yang, Global phase-portraits of plane homogeneous polynomial vector fields and stability of the origin, Systems Sci. Math. Sci., 10 (1997), 33-40. 

[34]

Y. Ye et al., Theory of Limit Cycles, vol. 66 of Transl. Math. Monographs, Amer. Math. Soc, Providence, RI, 1986.

show all references

References:
[1]

M. ÁlvarezA. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416.

[2]

A. A. Andronov, A. A. Vitt and S. E. Khaikin, Theory of Oscillators: Adiwes International Series in Physics, vol. 4, Elsevier, 2013.

[3]

J. C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations, 107 (1994), 80-95.  doi: 10.1006/jdeq.1994.1004.

[4]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields. Ⅰ, Nonlinear Anal., 29 (1997), 783-811.  doi: 10.1016/S0362-546X(96)00088-0.

[5]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields Ⅱ, Nonlinear Anal., 39 (2000), 351-363.  doi: 10.1016/S0362-546X(98)00177-1.

[6]

L. Cairó and J. Llibre, Phase portraits of planar semi-homogeneous vector fields (Ⅲ), Qual. Theory Dyn. Syst., 10 (2011), 203-246.  doi: 10.1007/s12346-011-0052-y.

[7]

A. CimaA. Gasull and F. Mañosas, On polynomial Hamiltonian planar vector fields, J. Differential Equations, 106 (1993), 367-383.  doi: 10.1006/jdeq.1993.1112.

[8]

A. Cima and J. Llibre, Algebraic and topological classification of the homogeneous cubic vector fields in the plane, J. Math. Anal. Appl., 147 (1990), 420-448.  doi: 10.1016/0022-247X(90)90359-N.

[9]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 257 (2014), 1623-1661.  doi: 10.1016/j.jde.2014.05.024.

[10]

I. E. ColakJ. Llibre and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Adv. Math., 259 (2014), 655-687.  doi: 10.1016/j.aim.2014.04.002.

[11]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 258 (2015), 846-879.  doi: 10.1016/j.jde.2014.10.006.

[12]

I. E. ColakJ. Llibre and C. Valls, Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equations, 262 (2017), 5518-5533.  doi: 10.1016/j.jde.2017.02.001.

[13]

F. S. DiasJ. Llibre and C. Valls, Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers, Math. Comput. Simulation, 144 (2018), 60-77.  doi: 10.1016/j.matcom.2017.06.002.

[14]

F. Dumortier, Techniques in the theory of local bifurcations: Blow-up, normal forms, nilpotent bifurcations, singular perturbations, in Bifurcations and Periodic Orbits of Vector Fields, Springer, 1993, 19–73.

[15]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, Berlin, 2006.

[16]

A. GasullA. Guillamon and V. Mañosa, Phase portrait of Hamiltonian systems with homogeneous nonlinearities, Nonlinear Anal., 42 (2000), 679-707.  doi: 10.1016/S0362-546X(99)00131-5.

[17]

H. Goldstein, Classical Mechanics, Addison-Wesley Press, Inc., Cambridge, Mass, 1951.

[18]

A. Guillamon and C. Pantazi, Phase portraits of separable Hamiltonian systems, Nonlinear Anal., 74 (2011), 4012-4035.  doi: 10.1016/j.na.2011.03.030.

[19]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4757-2421-9.

[20]

J. S. W. Lamb and M. Roberts, Reversible equivariant linear systems, J. Differential Equations, 159 (1999), 239-279.  doi: 10.1006/jdeq.1999.3632.

[21]

H. LiangJ. Huang and Y. Zhao, Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems, Nonlinear Dynam., 78 (2014), 1659-1681.  doi: 10.1007/s11071-014-1541-8.

[22]

J. LlibreY. P. Martínez and C. Vidal, Phase portraits of linear type centers of polynomial hamiltonian systems with hamiltonian function of degree $5$ of the form $H = H_1(x)+ H_2(y)$, Discrete Contin. Dyn. Syst. Ser., 39 (2019), 75-113.  doi: 10.3934/dcds.2019004.

[23]

J. LlibreY. P. Martínez and C. Vidal, Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the $y$-axis, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 887-912.  doi: 10.3934/dcdsb.2018047.

[24]

J. LlibreR. Oliveira and C. Valls, Phase portraits for some symmetric Riccati cubic polynomial differential equations, Topology Appl., 234 (2018), 220-237.  doi: 10.1016/j.topol.2017.11.023.

[25]

J. Llibre and C. Pessoa, Phase portraits for quadratic homogeneous polynomial vector fields on $\Bbb S^2$, Rend. Circ. Mat. Palermo, 58 (2009), 361-406.  doi: 10.1007/s12215-009-0030-2.

[26]

N. Minorsky, Nonlinear Oscillations, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962.

[27]

D. A. Neumann, Classification of continuous flows on $2$-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73-81.  doi: 10.1090/S0002-9939-1975-0356138-6.

[28]

H. Poincaré, Mémoire sur les courbes définies par une équation différentielle (i), Journal de Mathématiques Pures et Appliquées, 7 (1881), 375–422.

[29]

B. Qiu and H. Liang, Classification of global phase portrait of planar quintic quasi-homogeneous coprime polynomial systems, Qual. Theory Dyn. Syst., 16 (2017), 417-451.  doi: 10.1007/s12346-016-0199-7.

[30]

J. Reyn, Phase Portraits of Planar Quadratic Systems, vol. 583, Springer, New York, 2007.

[31]

D. Schlomiuk and X. Zhang, Quadratic differential systems with complex conjugate invariant lines meeting at a finite point, J. Differential Equations, 265 (2018), 3650-3684.  doi: 10.1016/j.jde.2018.05.014.

[32]

Y. Tian and Y. Zhao, Global phase portraits and bifurcation diagrams for Hamiltonian systems of linear plus quartic homogeneous polynomials symmetric with respect to the $y$-axis, Nonlinear Anal., 192 (2020), 111658, 27pp. doi: 10.1016/j.na.2019.111658.

[33]

X. Yang, Global phase-portraits of plane homogeneous polynomial vector fields and stability of the origin, Systems Sci. Math. Sci., 10 (1997), 33-40. 

[34]

Y. Ye et al., Theory of Limit Cycles, vol. 66 of Transl. Math. Monographs, Amer. Math. Soc, Providence, RI, 1986.

Figure 1.  Phase portraits of system (3)
Figure 2.  Bifurcation diagram of system $\left({{\bf{I}}{\bf{.5}}} \right)$
Figure 3.  The local phase portrait of the system (8) at the origin
Figure 4.  Local phase portrait of system $\left({{\bf{I}}{\bf{.1}}} \right)$ on the Poincaré disk
Figure 5.  The local phase portrait of system (13) at origin for $ c = 0 $
Figure 6.  The local phase portrait of the system (14) at the origin
Figure 7.  All the local phase portraits of system $\left({{\bf{I}}{\bf{.2}}} \right)$ on the Poincaré disk
Figure 8.  The local phase portrait of the system (15) at the origin for $ a\leq0 $
Figure 9.  All the local phase portraits of system $\left({{\bf{I}}{\bf{.3}}} \right)$ on the Poincaré disk
Figure 10.  The local phase portraits of system (17) at $ p_1^{\pm} $ for $ a<0 $
Figure 11.  The local phase portraits of system $\left({{\bf{I}}{\bf{.4}}} \right)$ on the Poincaré disk
Figure 12.  The local phase portraits of system $\left({{\bf{I}}{\bf{.5}}} \right)$ with $ \Delta>0 $ on the Poincaré disk
Table 1.  Algebraic classification of system (6)
$b$ $\omega(z)$ = 0ConditionsRoots of $\omega(z)$Linear change Normal forms
$b = 0$ Linear equation $a = 0, c\neq0$No roots $\left(x, y\right)\mapsto\left(c^{-1/3}x, c^{-1/3}y\right)$ $({\bf{I.1}})$
$a\neq0, c\in\mathbb{R}$One root $\left(x, y\right)\mapsto\left(a^{-1/3}x, a^{-1/3}y\right)$ $({\bf{I.2}})$
$b\neq0$ Quadratic equation $a\in\mathbb{R}, c = 0$ $0$ is a root $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.3}})$
$a\in\mathbb{R}, c\neq0, \Delta = 0$Multiple root $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.4}})$
$a\in\mathbb{R}, c\neq0, \Delta\neq0$Two simple roots $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.5}})$
$b$ $\omega(z)$ = 0ConditionsRoots of $\omega(z)$Linear change Normal forms
$b = 0$ Linear equation $a = 0, c\neq0$No roots $\left(x, y\right)\mapsto\left(c^{-1/3}x, c^{-1/3}y\right)$ $({\bf{I.1}})$
$a\neq0, c\in\mathbb{R}$One root $\left(x, y\right)\mapsto\left(a^{-1/3}x, a^{-1/3}y\right)$ $({\bf{I.2}})$
$b\neq0$ Quadratic equation $a\in\mathbb{R}, c = 0$ $0$ is a root $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.3}})$
$a\in\mathbb{R}, c\neq0, \Delta = 0$Multiple root $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.4}})$
$a\in\mathbb{R}, c\neq0, \Delta\neq0$Two simple roots $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.5}})$
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